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Full text of "Analytical Mechanics"

r
The equation -rf£ = -&2sin# is at-
aot integrable in a finite number D£ terms; therefore the solution of equation (X) must be given either in an approximate form, or it must be expressed as an infinite series. FIRST APPROXIMATION. —When 9 is small sin 0 may be replaced by 9. Therefore we can write
if- -
(X')
G. 137
or
dt
(X")
where c2 =
mgD
It will be observed that the last two
equations are of the same type as equations (I7) and (II7) of p. 297; the differential equations of simple harmonic motion. Therefore the motion of the physical pendulum is approximately harmonic. Hence we can apply to the present problem the results which were obtained in discussing simple harmonic motion. Thus the expression for the displacement is                      0=«sin («*+«),                          (XI)
where a is the amplitude, i.e., the maximum angular displacement of the pendulum. On the other hand the period of the pendulum is            2 ^
=2r\44
mgD
z + mD2 mgD
(XII)
(XIII)